Can we trust in numerical computations of chaotic
solutions of dynamical systems ?
René Lozi
To cite this version:
René Lozi. Can we trust in numerical computations of chaotic solutions of dynamical systems ?. World Scientific Series on Nonlinear Science, World Scientific, 2013,
Topology and Dynamics of Chaos In Celebration of Robert Gilmore’s 70th Birthday, 84, pp.63-98. <http://www.worldscientific.com/doi/abs/10.1142/9789814434867 0004>.
<10.1142/9789814434867 0004>. <hal-00682818>
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CAN WE TRUST IN NUMERICAL COMPUTATIONS OF
CHAOTIC SOLUTIONS OF DYNAMICAL SYSTEMS ?
R. LOZI
Laboratoire J.A. Dieudonné - UMR CNRS 7351
Université de Nice Sophia-Antipolis
Parc Valrose
06108 NICE Cedex 02
FRANCE
E-mail: [email protected]
Since the famous paper of E. Lorenz in 1963 numerical computations using computers
play a central role in order to display and analyze solutions of nonlinear dynamical
systems. By these means new structures have been emphasized like hyperbolic and/or
strange attractors. However theoretical proofs of their existence are very di¢ cult and
limited to very special linear cases. Computer aided proofs are also complex and require
special interval arithmetic analysis. Nevertheless, numerous researchers in several …elds
linked to chaotic dynamical systems are con…dent in the numerical solutions they found
using popular software and publish without checking carefully the reliability of their
results. In the simple case of discrete dynamical systems (e.g. Hénon map) there are
concerns about the nature of what a computer …nd out : long unstable pseudo-orbits or
strange attractors? The shadowing property and its generalizations which ensure that
pseudo-orbits of a homeomorphism can be traceable by actual orbits even if rounding
errors are not inevitable are not of great help in order to validate the numerical results.
Continuous dynamical systems (e.g. Chua, Lorenz, Rössler) are even more di¢ cult to
handle in this scope and researchers have to be very cautious to back up theory with
numerical computations. We present a survey of the topic based on these, only few, but
well studied models.
To appear in: “From Laser Dynamics to Topology of Chaos,” (celebrating the
70th birthday of Prof Robert Gilmore, Rouen, June 28-30, 2011), Ed. Ch. Letellier.
1. Introduction
Since the famous paper of E. Lorenz in 19631 , numerical computations using computers play a central role in order to display and analyze solutions of nonlinear
dynamical systems. By these means new structures have been emphasized like hyperbolic and/or strange attractors. However theoretical proofs of their existence
are very di¢ cult and limited to very special linear cases.2 Computer aided proofs
are also complex and require special interval arithmetic analysis.3;4 Nevertheless,
numerous researchers in several …elds linked to chaotic dynamical systems are con…dent in the numerical solutions they found using popular software and publish
1
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without checking carefully the reliability of their results. In the simple case of discrete dynamical systems (e.g. Hénon map5 ) there are concerns about the nature
of what a computer …nd out : long unstable pseudo-orbits or strange attractors?6
The shadowing property which ensures that pseudo-orbits of a homeomorphism
can be traceable by actual orbits, even if rounding errors are not inevitable, is
not a great help in order to validate the numerical results.7 Continuous dynamical systems (e.g. Chua, Lorenz, Rössler) are even more di¢ cult to handle in this
scope and researchers have to be very cautious to back up theory with numerical
computations.8;9 We present a survey of the topic based on these, only few, but
well studied models.10
In Sec. 2 we de…ne the paradigm for possibly ‡awed computations: continuous and discrete chaotic dissipative dynamical systems. In Sec. 3, some undesirable collapsing e¤ects are highligted for an example of strange attractor and for
1-dimensional discrete dynamical systems (logistic and tent maps). The shadowing properties (parameter-shifted, orbit-shifted shadowing) are presented in Sec. 4,
together with the classical mappings of the plane into itself: the Hénon and Lozi
maps. Finally in Sec. 5 the case of the seminal Lorenz model and its following
metaphors: Rössler and Chua equations is examined.
2. Continuous and discrete chaotic dissipative dynamical systems:
a paradigm for possibly ‡awed computations
2.1. Some classes of dynamical system
Dynamical systems are involved in the modeling of phenomena which evolve in time.
Their theory attempts to understand, or at least describe in form of mathematical
equations, the changes over time that occur in biological, chimical, economical,
…nancial, electronical, physical or arti…cial systems. Examples of such systems
include the long-term behavior of solar system (sun and planets) or galaxies, the
weather, the growth of crystals, the struggle for life between competiting species,
the stock market, the formation of tra¢ c jams, etc.
Dynamical systems can be continuous vs discrete, autonomous vs nonautonomous, conservative vs dissipative, linear vs nonlinear, etc. When di¤erential
equations are used, the theory of dynamical systems is called continuous dynamical
systems. Instead when di¤erence equations are employed the theory is called discrete dynamical systems. Some situations may also be modeled by mixed operators
such as di¤erential-di¤erence equations. It is the theory of hybrid systems.
In mathematics, an autonomous system or autonomous di¤erential equation is
a system of ordinary di¤erential equations which does not explicitly depend on the
independent variable, if it is not the case the system is called non-autonomous.
Classical mechanics deals with dynamical systems without damping or friction
as the ideal pendula, the solar system. In this case the dynamical systems involved
are conservative dynamical systems. When damping or friction occur the dynamical
systems are dissipative.
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Linear dynamical systems can be solved in terms of simple functions and the
behavior of all orbits classi…ed, on the contrary there are in general no explicit
solutions of non linear dynamical systems which model more complex phenomena.
Before the advent of fast computing machines, solving a non linear dynamical
system required sophisticated mathematical techniques and could be accomplished
only for a small class of dynamical systems. Numerical methods implemented on
computers have simpli…ed the task of determining the orbits of a dynamical system.
However when chaotic dynamical systems are studied the crucial question is: can
we rely on these solutions ?
Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an e¤ect which is popularly referred to as the butter‡y
e¤ect. Small di¤erences in initial conditions (such as those due to rounding errors in
numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general. This happens even though these
systems are deterministic, meaning that their future behavior is fully determined
by their initial conditions, with no random elements involved. In other words, the
deterministic nature of these systems does not make them predictable. The …rst
example of such chaotic continuous system in the dissipative case was pointed out
by the meteorologist E. Lorenz in 1963.1
In this article we limit our study to chaotic (hence non linear) dissipative dynamical systems, either continuous or discrete, autonomous or non-autonomous.
The case of linear system is not relevant because most of the solutions are given
by closed formulas. The case of conservative system is much more di¢ cult to handle as the lack of friction (dissipation of energy) leads to exponential increasing of
rounding errors. Dedicated techniques are necessary to obtain reliable solution.11
Although there exist peculiar mathematical tools in order to study nonautonomous dynamical systems, they can be easily transformed in autonomous
systems increasing by one the dimension of the space variable. Then we only consider autonomous sytems. We focus our study to the most popular models: for
the discrete case: logistic and tent map in 1-dimension, Hénon and Lozi map in
2-dimension; for the continuous case: Lorenz, Rössler and Chua model.
2.2. Poincaré map: a bridge between continuous and discrete
dynamical system
Generally (i.e. when there exist a periodic solution) the Poincaré map allows us
to build a correspondence between continuous and discrete dynamical systems. If
we consider a 3-dimensional continuous dynamical system (i.e. a system of three
di¤erential autonomous equations):
8 :
< x1 = f1 (x1 ; x2 ; x3 )
:
x = f2 (x1 ; x2 ; x3 )
: :2
x3 = f3 (x1 ; x2 ; x3 )
(1)
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Figure 1.
Solutions of a di¤erential system in R3 .
the solution of such a system can be seen as a parametric curve (x1 (t) ; x2 (t) ; x3 (t))
in the space R3 . A periodic solution (also called a cycle) is no more than a loop in
this space as shown on Fig. 1 for the solution starting from, and arriving to, the
same initial condition X0 .
Poincaré map de…ned in a neighborhood of this cycle is the map ' of the plane
= R2 into itself which associates to the initial point belonging to this plane, the
…rst return point of the solution starting from this very initial point ' : X 2 !
' (X) 2 (e.g. on Fig. 1. the …rst intersection point X2 of the plane with the
solution starting from X1 , ' : X1 2 ! ' (X1 ) = X2 2 ). Then the study of
n-dimensional continuous system is equivalent to the study of (n 1)-dimensional
discrete system.
Figs. 2 and 3 display the discrete periodic orbit fX0 ; X1 ; X2 ; X3 ; X4 = X0 g
associated to the continuous periodic orbit of period 4:
'(4) (X0 ) = ' ' ' ' (X0 ) = X0
3. Collapsing e¤ects
3.1. Undesirable chaotic transient
In 2008, Z. Elhadj and J. C. Sprott12 introduced a two-dimensional discrete mapping
with C 1 multifold chaotic attractors. They studied a modifed Hénon map given
by:
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Figure 2.
Discrete periodic orbit associated to the continuous periodic orbit of period 4.
Figure 3.
f (xn ; yn ) =
Continuous period 4 orbit.
xn+1
yn+1
=
1
a sin (xn ) + byn
xn
(2)
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where the quadratic term x2 in the Hénon map (see Sec. 4) is replaced by the
nonlinear term sin x. They studied this model for all values of a and b. The
essential motivation for this work was to develop a C 1 mapping that is capable
of generating chaotic attractors with “multifolds”via a period-doubling bifurcation
route to chaos which has not been studied before in the literature.
They prove the following theorem on the exitence of bounded orbits
Theorem 3.1. (Elhadj & Sprott) The orbits of the map (2) are bounded for all
a 2 R, and jbj < 1, and all initial conditions (x0 ; x1 ) 2 R2 .
The existence of chaotic attractors is only inferred numerically. They display
four examples, convincing at …rst glance, of what they call “chaotic multifold attractors”. Unfortunately one of it obtained for a = 4:0 and b = 0:9 (Fig. 4) is no more
than a long transient regime which collapses to a trivial and degenerate behavior:
a periodic orbit of period 6 when the computation is done for su¢ cently large value
of n. When programming in Language C (Borland R compiler), using a computer
with Intel DuoCore2 processor and computing with double precision numbers, from
any initial points after a transient regime for approximatively 110; 000 iterates (actually the length of the transient regime depends upon the initial value) the orbit
is trapped to the period-6 attractor given by:
x120003 = 8:95855079898761453 = x120009 = x120015 =
,
x120004 = 10:96940289429559630 = x120010 = x120016 =
,
x120005 = 13:06132591362670500 = x120011 =
x120006 = 8:97249334266406962 =
,
x120007 = 11:90071070225514802 =
,
x120008 = 13:07497800339342220 =
.
,
This attractor is showed on Fig. 5.
Remark 3.1. It is obvious that the phase space (xn ; xn+1 ) on which the points
(xn ; xn+1 ) 2 R2 are computed is …nite when …nite arithmetic replaces continum
state spaces and one can object that every orbit of a mapping must be periodic
in this …nite space. However with double precision numbers for each component,
it is generally possible (in presence of chaotic attractor) to obtain periodic orbit
with period as long as 1011 (see Lozi map in Sec. 4.2) which could be a good
approximation for the attractor. Instead an attracting period-6 orbit has a very
di¤erent behavior than a strange attractor because such orbit does not possesses
the sensitivity to initial condition property, which a minimal necessary condition
(but not a su¢ cient one) of existence of chaos.
This example of long chaotic transient regime hiding a periodic attractor with
a very short period is not unique in scienti…c literature, numerous researchers in
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Figure 4. Chaotic multifold attractor of the map (2) for a = 4:0 and b = 0:9.
Sprott12 ).
(Elhadj &
Figure 5. Period 6 attracting orbit of the map (2) for a = 4:0 and b = 0:9 (Elhadj & Sprott12 ).
The six points on the graph are magni…ed.
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several …elds linked to chaotic dynamical systems are con…dent in the numerical
solutions they found using popular softwares and publish without checking carefully
the reliability of their results. Most of the time they compute only few iterates (i.e.
few means less than 109 ) of mapping and falsely conclude the existence of chaotic
regimes upon these numerical clues.
3.2. Enigmatic computations for the logistic map (1838)
In 1838 the belgium mathematician Pierre François Verhulst13 introduced a di¤erential equation modelling the grow of population in a simple demographic model,
as an improvement of the Malthusian growth model, in which some resistence to
the natural increase of population is added.
dp
dt
= mp np2
p (0) = p0
(3)
the function p(t) being the size of the population of the mankind. He latter, in
184514 , called logistic function the solution of this equation. Putting
n
(4)
x (t) = p (t)
m
Eq. 3 is equivalent to
dx
= mx (1 x)
(5)
dt
In 1973, the biologist Sir R. M. May introduced the nonlinear, discrete time
dynamical system
xn+1 = rxn (1
xn )
(6)
as a model for the ‡uctuations in the population of fruit ‡ies in a closed container
with constant food.15 Due to the similarity of both equations although one is a
continuous dynamical system, and the other a discrete one, he called Eq. 6 logistic
equation. The logistic map fr : [0; 1] ! [0; 1]
fr (x) = rx (1
x)
(7)
associated to Eq. 6 and generally considered for r 2 [0; 4] is often cited as an
archetypal example of how complex, chaotic behaviour can arise from very simple
non-linear dynamical equations.
This dynamical system which has excellent ergodic properties on the real interval
[0; 1] has been extensively studied especially by R. M. May16 , and J. Feigenbaum17
who introduced what is now called the Feigenbaum’s constant
= 4:669201609102990671853203820466201617258185577:::
explaining by a new theory (period doubling bifurcation) the onset of chaos.
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For every value of r there exist two …xed points: x = 0 which is always unstable
and x = r r 1 which is stable for r 2 ]1; 3[ and unstable for r 2 ]0; 1[ [ ]1; 4[ (see Fig.
6).
When r = 4 , the system is chaotic. The set
(
p
p )
5
5 5+ 5
;
= f0:3454915028; 0:9045084972g
8
8
is the period-2 orbit. In fact there exist in…nity of periodic orbits and in…nity of
periods (furthermore several distinct periodic orbits having the same period can
coexist). This dynamical system possesses an invariant measure
(x) =
1
p
x (1
x)
(see Fig. 7).
It is quite surprising that a simple quadratic equation can exhibit such complex
behaviour. If the logistic equation with r = 4 modelled the growth of fruit ‡ies, then
their population would exhibit erratic ‡uctuations from one generation to another.
In the coordinate sytem
X = 2x
Y = 2y
1
1
(8)
the map of Eq. 7 written in the equivalent form (for r = 4)
g (X) = 1
2X 2
(9)
was studied in the interval [ 1; 1] by T. S. Ulam and J. von Neumann well before
the modern chaos era. They proposed it as a computer random number generator.18
In order to compute periodic orbits whose period is longer than 2 the use of
computer is required, as it is equivalent to …nd roots of polynomial equation of
degree greater than 4, for which Galois theory teaches that no closed formula is
available. However, numerical computation on computer uses ordinarily double
precision numbers (IEEE-754) so that the working interval contains roughly 1016
representable points. Doing such a computation19 in Eq. 6 with r = 4 with 1,000
randomly chosen initial guesses, 596, i.e., the majority, converge to the unstable
…xed point x = 0, and 404 converge to a cycle of period 15,784,521 (see Table 1a).
In an experimental work,20 O. E. Lanford III, does the same search of numerical
periodic solution of the logistic equation under the form of Eq. 9. The precise
discretization studied is obtained exploiting evenness of this equation to fold the
interval [ 1; 0] to [0; 1], i.e. replacing Eq. 9 by
G (X) = 1
2X 2
(10)
on [0; 1]. It is not di¢ cult to see that the folded mapping has the same set of
periods as the original one. In order to avoid the particular discretization of this
interval when the standard IEEE-754 is used for double precision numbers, the
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Table 1a. Coexisting periodic orbits with 1,000 initial guesses
Period
Orbit
Relative Basin Size
1
f0g (unstable …xed point)
596 over 1; 000
15; 784; 521 Scattered over the interval
404 over 1; 000
Table 1b. Coexisting periodic orbits with 1,000 initial guesses
Period
Orbit
Relative Basin Size
1
f0g (unstable …xed point)
890 over 1; 000
1; 107; 319 Scattered over the interval
2 over 1; 000
3; 490; 273 Scattered over the interval
108 over 1; 000
working interval is then shifted from [0; 1] to [1; 2] by translation, and the iteration
of the translated folded map is programmed in straightforward way. Out of 1,000
randomly chosen initial points, 890, i.e., the overwhelming majority, converged to
the …xed point corresponding to the unstable …xed point x = 1 in the original
representation of Eq. 9, 108 converged to a cycle of period 3,490,273, the remaining
2 converged to a cycle of period 1,107,319 (see Table 1b).
Thus, in both cases at least, the very long-term behaviour of numerical orbits
is, for a substantial fraction of initial points, in ‡agrant disagreement with the true
behaviour of typical orbits of the original smooth logistic map.
In others numerical experiments we have performed,21;22 the computer working
with …xed …nite precision is able to represent …nitely many points in the interval in
question. It is probably good, for purposes of orientation, to think of the case where
the representable points are uniformly spaced in the interval. The true logistic map
is then approximated by a discretized map, sending the …nite set of representable
points in the interval to itself.
Describing the discretized mapping exactly is usually complicated, but it is
roughly the mapping obtained by applying the exact smooth mapping to each of the
discrete representable points and “rounding”the result to the nearest representable
point. In our experiments, uniformly spaced points in the interval with several order
of discretization (ranging from 9 to 2,001 points) are involved. In each experiment
the questions addressed are:
how many periodic cycles are there, and what are their periods ?
how large are their respective basins of attraction, i.e., for each periodic
cycle, how many initial points give orbits with eventually land on the cycle
in question ?
Table 2 shows coexisting periodic orbits for the discretization with regular
meshes of N = 9, 10 and 11 points. There are respectively 3, 2 and 2 cycles.
Table 3 displays cases N = 99, 100 and 101 points, there are exactly 2, 4 and 3
cycles. Table 4 stands for regular meshes of N = 1999, N = 2000 and N = 2001
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Figure 6.
Cobweb diagram of the logistic map f4 (x).
points.
It seems that the computation of numerical approximations of the periodic orbits
leads to unpredictable and somewhat enigmatic results. As says O. E. Lanford III,20
“The reason is that because of the expansivity of the mapping the growth of roundo¤
error normally means that the computed orbit will remain near the true orbit with
the chosen initial condition only for a relatively small number of steps typically of
the order of the number of bits of precision with which the calculation is done It
is true that the above mapping like many ‘chaotic’mappings satis…es a shadowing
theorem (see Sec. 4.3 in this article) which ensures that the computed orbit stays
near to some true orbit over arbitrarily large numbers of steps. The ‡aw in this idea
as an explanation of the behavior of computed orbits is that the shadowing theorem
says that the computed orbit approximates some true orbit but not necessarily that
it approximates a typical one.”
He adds, “This suggests the discouraging possibility that this problem may be
as hard of that of non equilibrium statistical mechanics”
The existence of very short periodic orbit (Tables 1a, 1b), the existence of a non
constant invariant measure (Fig. 7) an the easily recognized shape of the function
in the phase space avoid the use of the logistic map as a Pseudo Random Number
Generator. However, its very simple implementation in computer program led some
authors to use it as a base of cryptosytem.23;24
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Figure 7.
Invariant measure
(x) =
p
1
x(1 x)
of the logistic map f4 (x).
Table 2. Computation on regular meshes of N points
N Period
Orbit
Basin Size
9
1
f0g
3 over 9
9
1
f6g
2 over 9
9
1
f3; 7g
4 over 9
10
1
f0g
2 over 10
10
2
f3; 8g
8 over 11
11
1
f0g
3 over 11
11
4
f3; 8; 6; 9g
8 over 11
3.3. Collapsing orbit of the symmetric tent map
Another often studied discrete dynamical system is de…ned by the symmetric tent
map on the interval J = [ 1; 1]
xn+1 = Ta (xn )
Ta (x) = 1
a jxj
(11)
(12)
Despite its simple shape (see Fig. 8), it has several interesting properties. First,
when the parameter value a = 2, the system possesses chaotic orbits. Because of
its piecewise-linear structure, it is easy to …nd those orbits explicitly. More, owing
to its simple de…nition, the symmetric tent map’s shape under iteration is very well
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Table 3. Computation on regular meshes of 99, 100 and 101 points
N
Period
Orbit
Basin Size
99
1
f0g
3 over 99
99
10
f3; 11; 39; 93; 18; 58; 94; 15; 50; 97g
96 over 99
100
1
f0g
2 over 100
100
1
f74g
2 over 100
100
6
f11; 39; 94; 18; 58; 96g
72 over 100
100
7
f7; 26; 76; 70; 82; 56; 97g
24 over 100
101
1
f0g
3 over 101
101
1
f75g
2 over 101
101
1
f19; 61; 95g
96 over 101
Table 4. Computation on regular meshes of 1; 999, 2; 000
N
Period
Orbit
1999
1
f0g
1999
4
f554; 1601; 1272; 1848g
1999
8
f3; 11; 43; 168; 615; 1702; 1008; 1997g
2000
1
f0g
2000
1
f1499g
2000
2
f691; 1808g
2000
3
f376; 1221; 1900g
2000
8
f3; 11; 43; 168; 615; 1703; 1008; 1998g
2001
1
f0g
2001
1
f1500g
2001
2
f691; 1809g
2001
8
f3; 11; 43; 168; 615; 1703; 1011; 1999g
2001
18
f35; 137; 510; 1519; 1461; 1574; g
2001
25
f27; 106; 401; 1282; 1840; 588; g
and 2; 001 points
Basin Size
3 over 1999
990 over 1999
1006 over 1999
2 over 2000
14 over 2000
138 over 2000
6 over 2000
1840 over 2000
5 over 2001
34 over 2001
92 over 2001
608 over 2001
263 over 2001
1262 over 2001
understood. The invariant measure is the Lebesgue measure. Finally, and perhaps
the most important, the tent map is conjugate to the logistic map, which in turn is
conjugate to the Hénon map (see Sec. 4.1) for small values of b.25
However the symmetric tent map is dramatically numerically instable:
Sharkovski¼¬’s theorem applies for it.26 When a = 2 there exists a period three
orbit, which implies that there is in…nity of periodic orbits. Nevertheless the orbit
of almost every point of the interval J of the discretized tent map eventually wind
up to the (unstable) …xed point x = 1 (this is due to the binary structure of
‡oating points) and there is no numerical attracting periodic orbit.27
The numerical behaviour of iterates with respect to chaos is worse than the
numerical behaviour of iterates of the approximated logistic map. This is why the
tent map is never used to generate numerically chaotic numbers.
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Figure 8.
Tent map T2 (x).
3.4. statistical properties
Many others examples could be given, but those given may serve to illustrate the
intriguing character of the results, the outcomes proves to be extremely sensitive
to the details of the experiment, but the …ndings all have a similar ‡avour: a
relatively small number of cycles attract near all orbits, and the lengths of these
signi…cant cycles are much larger than one but much smaller than the number of
representable points. P. Diamond and al.28;29 , suggest that statistical properties of
the phenomenon of computational collapse of discretized chaotic mapping can be
modelled by random mappings with an absorbing centre. The model gives results
which are very much in line with computational experiments and there appears to
be a type of universality summarised by an Arcsin law. The e¤ects are discussed
with special reference to the family of dynamical systems
xn+1 = 1
j1
2xn j ; 0 6 x 6 1; 1 6 l 6 2
l
(13)
Computer experiments show close agreement with prediction of the model. However these results are of statistical nature, they do not give accurate information
on the exact nature of the periodic orbits (e.g. length of the shortest or the greatest one, size of their basin of attraction ...). Following this work, G. Yuan and J.
A. Yorke30 study precisely the collapse to the repelling …xed point x = 1 of the
iterates of the one dimensional dynamical system
xn+1 = 1
l
2 jxn j ;
1 6 x 6 1; l > 2
(14)
for a large fraction of initial conditions when calculated on a mesh of N points
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equally spaced from
to this system
1 to +1 (as we have done in Sec. 3.2). The map associated
l
Ta;l (x) = 1
a jxj
being an usual nonlinear generalization of the map Ta (x) of Eq. 12. They not only
prove rigorously that the collapsing e¤ect does not vanish when arbitrarily high
numerical precision is employed, but also they give a lower bound of the probability
for which it happens. They de…ne Pcollapse which is the probability that there exists
n such that xn = 1 and they give the proof that Pcollapse depends only on the
numbers N and l. The lower bound of Pcollapse is given by
lim inf Pcollapse >
N !1
p
K 0 = lK
K=
2
K 0 [1
1
1
X
i=1
h
2
ki2
erf (K 0 )] exp (K 0 ) > 0
1=:2 1=l
!1=2
1=l
ki = l (i + 1=2)
with
1=l
1 + (2l)
2
erf(x) = p
(i
Z
x
e
1=l
1=2)
t2
(15)
2
1=2
1
i
dt
(16)
0
They plot the curve given by Eq. 15 along with the numerical results (Figure
9). Each numerical datum is obtained as follows. For each l they sample 10; 000
pairs of l; x from(l 0:01l; l + 0:01l) ( 1; 1)with uniform distribution. For each
sample they keep iterating the dynamical system de…ned by Eq. 14 with the initial
condition x until the numerical trajectory repeats. Then they calculate the portion
of trajectories that eventually map to 1. The deviation is clear since Eq. 15 only
gives a lower bound. Nonetheless the theoretical curve reveals the fact that Pcollapse
is already substantial for l = 3 and it predicts that Pcollapse increases as l increases
and that liml!1 Pcollapse = 1.
4. Shadowing and parameter-shifted shadowing property of
mappings of the plane
4.1. Hénon map (1976) found by mistake
The chaotic behavior of mapping of the real line is relatively simple compared to
the behavior of mapping on the plane. These mappings could be seen as a simple
expansion in a phase space with increased dimension of both models (logistic and
tent) we have introduced in the previous section. However they have been discovered
not in this “ascending” way, instead they come in “descending way” as metaphor
of Poincaré map of 3-D continuous dynamical systems.
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Figure 9. Pcollapse as a function of l. The curve in this …gure is the lower bound computed from
Eq. 15. Numerical results are obtained by using di¤erent numerical precisions are also shown in
this …gure "+" -double precision " " -single precision " "-…xed precision 10 12 (from 30 ).
Figure 10.
Observatory of Nice, the o¢ ce of Michel Hénon was located inside the building.
In order to study numerically the properties of the Lorenz attractor1 , M. Hénon
an astronom of the observatory of Nice, France (see Fig. 10) introduced in 19765 a
simpli…ed model of the Poincaré map of this attractor. The Lorenz attractor being
in dimension 3, the corresponding Poincaré map is a map from the plane R2 to R2 .
The Hénon map is then also de…ned in dimension 2 as
Ha;b :
x
y
=
y + 1 ax2
bx
(17)
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It is associated to the dynamical system
xn+1 = yn + 1 ax2n
yn+1 = bxn
(18)
For the parameter value a = 1:4, b = 0:3, M. Hénon pointed out numerically
that there exists an attractor with fractal structure. This was the …rst example
of strange attractor (previously introduced by D. Ruelle and F. Takens31 ) for a
mapping de…ned by an analytic formula.
As highlighted in the sequence of Figs. 11-14, the like-Cantor set structure in
one direction orthogonal to the invariant manifold in this simple mapping was a
dramatic surprise in the community of physicists and mathematicians.
Nowadays hundreds of research papers have been published on this prototypical map in order to fully understand its innermost structure. However as for 1dimensional dynamical systems, there is a discrepancy between the mathematical
properties of this map in the plane and the numerical computations done using
(IEEE-754) double precision numbers.
If we call Megaperiodic orbits,6 those whose length of the period belongs to the
interval of natural numbers 106 ; 109 and Gigaperiodic orbits, those whose length
of the period belongs to the interval 106 ; 109 , Hénon map possesses both Mega
and Gigaperiodic orbits. On a Dell computer with a Pentium IV microprocessor
running at 1.5 Gigahertz frequency, using a Borland C compiler and computing
with ordinary (IEEE-754) double precision numbers, one can …nd for a = 1:4 and
b = 0:3 one attracting period of length 3; 800; 716; 788, i.e. two hundred forty times
longer than the longest period of the one-dimensional logistic map (Table 1a). This
periodic orbit (we call it here Orbit 1) is numerically slowly attracting. Starting
with the initial value
(x0 ; y0 )1 = ( 0:35766; 0:14722356)
one winds up:
(x11;574;730;767;y11;574;730;767 )1 = (x15;375;447;555;y15;375;447;555 )1
= (1:27297361350955662; 0:0115735710153616837)
The length of this period is obtained subtracting
15; 375; 447; 555
11; 574; 730; 767 = 3; 800; 716; 788
However this Gigaperiodic orbit is not unique: starting with the other initial value
(x0 ; y0 )2 = (0:4725166; 0:25112222222356)
the following Megaperiodic orbit (Orbit 2) of period 310; 946; 608 is computed
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(x12;935;492;515;y12;935;492;515 )2 = (x13;246;439;123;y13;246;439;123 )2
= (1:27297361350865113; 0:0115734779561870744)
Remark 4.1. This second orbit can be reached more rapidly starting form the
other initial value
(x0 ; y0 ) = (0:881877775591; 0:0000322222356)
then
(x4;459;790;707 ; y4;459;790;707 ) = (1:27297361350865113; 0:0115734779561870744)
Remark 4.2. It is possible that some others periodic orbits coexist with both Orbit
1 and Orbit 2. However there is no peculiar method but the brute force to …nd it
if any.
Remark 4.3. The comparison between Orbit 1 and Orbit 2 gives a perfect idea of
the sensitive dependence on initial conditions of chaotic attractors:
Orbit 1 passes through the point
(1:27297361350955662; 0:0115735710153616837)
and Orbit 2 passes through the point
(1:27297361350865113; 0:0115734779561870744)
The same digits of the coordinates of these points are bold printed, they are very
close.
The discovery of a strange attractor for maps of the plane boosted drastically
the research on chaos in the years 70’. This is miraculous considering that if M.
Hénon tried to test rigorously his model nowadays with powerful computers he
should only found long periodic orbits. In 1976, M. Hénon used the electronic
pocket calculator built by Hewlett-Packard company HP-65 (Fig. 15) and one of
the only two computers available at the university of Nice a IBM 7040 (Fig.16), the
other one was a IBM 1130, slower. The HP-65 …rst introduced in 1974 was worth
750 e (equivalent to 3; 500 e nowadays), the IBM 7040, was worth 1,100,000 e
(equivalent to 7; 800; 000 e in the year 2012).
He concluded,5 “The simple mapping (4) (i.e. Eq. 17 in this article) appears
to have the same basic properties as the Lorenz system. Its numerical exploration
is much simpler: in fact most of the exploratory work for the present paper was
carried out with a programmable pocket computer (HP-65). For the more extensive computations of Figures 2 to 6, we used a IBM 7040 computer, with 16-digit
accuracy. The solutions can be followed over a much longer time than in the case of
a system of di¤erential equations. The accuracy is also increased since there are no
integration errors. Lorenz (1963) inferred the Cantor-set structure of the attractor
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Figure 11.
Hénon strange attractor, 10000 successive points obtained by iteration of the mapping.
from reasoning, but could not observe it directly because the contracting ratio after
one ‘circuit’was too small: 7 10 5 . A similar experience was reported by Pomeau
(1976). In the present mapping, the contracting ratio after one iteration is 0:3, and
one can easily observe a number of successive levels in the hierarchy. This is also
facilitated by the larger number of points. Finally, for mathematical studies the
mapping (4) might also be easier to handle than a system of di¤erential equations.”
The large number of points was in fact very few5 : “The transversal structure
(across the curves) appears to be entirely di¤erent, and much more complex. Already on Figures 2 and 3 (i.e. Fig. 11 in this article) a number of curves can be
seen, and the visible thickness of some of them suggests that they have in fact an
underlying structure. Figure 4 (i.e. Fig. 12 in this article) is a magni…ed view of
the small square of Figure 3: some of the previous ‘curves’are indeed resolved now
into two or more components. The number n of iterations has been increased to
105 , in order to have a su¢ cient number of points in the small region examined.
The small square in Figure 4 is again magni…ed to produce Figure 5 (i.e. Fig. 13
in this article), with n increased to 106 : again the number of visible ‘curves’ increases. One more enlargement results in Fig. 6 (i.e. Fig. 14 in this article), with
n = 5 106 : the points become sparse but new curves can still easily be traced.”
In 1976, the one million e, IBM 7040 took several hours to compute …ve millions
points, today a basic three hundred euros laptop, runs the same computation in one
hundredth of a second.
With so few iterations nowadays the following claim5 : “These …gures strongly
suggest that the process of multiplication of ‘curves’will continue inde…nitely, and
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Figure 12. Enlargement of the squared region of Figure 11. The number of computed points is
increased to n = 105 .
that each apparent ‘curves’ is in fact made of an in…nity of quasi-parallel curves.
Moreover, Figures 4 to 6 indicate the existence of a hierarchical sequence of ‘levels’,
the structure being practically identical at each level save for a scale factor. This
is exactly the structure of a Cantor set.” should be obviously untrue. One can
assert that Hénon map was found not by chance (the reasoning of M. Hénon was
straightforward) but by mistake. It is a great luck for the expand of chaos studies!
4.2. Lozi map (1977) a tractable model
Using one of the …rst desktop electronic calculator HP-9820, he usually employed
for initiate to computer sciences, mathematics teacher trainees, R. Lozi found out,
on june 15, 1977, (on the scienti…c campus of the university of Nice, few kilometers apart of the observatory of this town of the French Riviera where M. Hénon
worked), that the linearized version of the Hénon map displayed numerically the
same structure of strange attractors, but the curves were replaced by straight lines.
He published this result in the proceedings of a conference on dynamical systems
which held in Nice in july of the same year,32 in a very short paper of one and an
half pages.
The aim of R. Lozi, a numerical analyst, was to allow algebraic computations
on an analog of the Hénon map, such direct computation being untractable on the
original model due to the square function. He changed this function with a absolute
value, de…ning the map
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Figure 13.
Figure 14.
Enlargement of the squared region of Figure 12, n = 106 .
Enlargement of the squared region of Figure 13, n = 5
La;b :
x
y
=
y + 1 a jxj
bx
106 .
(19)
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Figure 15.
Electronic pocket calculator HP-65.
Figure 16.
Computer IBM 7040.
and the associate dynamical system
xn+1 = yn + 1 a jxn j
yn+1 = bxn
(20)
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Figure 17.
Lozi map L1:7;0:5 .
He then found out on the plotter device attached to his small primitive computer
that for the value a = 1:7, b = 0:5, the same Cantor-set like structure was apparent.
Due to the linearity of this new model, it is possible to compute explicitly any
periodic orbit solving a linear system. Moreover in december 1979, M . Misiurewicz2
gave a talk in the famous conference organized in december 1979 at New-York by the
New-York academy of science in which he etablished rigorously the proof that what
he call “Lozi map” has a strange attractor for some parameter values (including
the genuine one a = 1:7, b = 0:5). Since that time, hundreds of papers have been
published on the topic.
Remark 4.4. The open set M of the parameter space (a; b) where the existence
of strange attractor is proved is now called the Misiurewicz domain. It is de…ned as
p
a2 1
0 < b < 1; a > b + 1; 2a + b < 4; a 2 > b + 2; b <
(2a + 1)
(21)
Today, in the same conditions of computation as for Hénon map (see Sec. 4.1),
running the computation during nineteen hours, one can …nd a Gigaperiodic attracting orbit of period 436; 170; 188; 959 more than one hundred times longer than
the period of Orbit 1 found for the Hénon map.
Starting with
(x0 ; y0 ) = (0:88187777591; 0:0000322222356)
one obtains
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(x686;295;403;186;y686;295;403;186 ) = (x250;125;214;227;y250;125;214;227 )
= (1:34348444450739479; 2:07670041497986548:10
There is a transient regime before the orbit is reached. It seems that there is
no periodic orbit with a smaller length. This could be due to the quasihyperbolic
nature of the attractor. However, the parameter-shifted shadowing property, the
orbit-shifted shadowing property of Lozi map (and generalized Lozi map), which
are new variations of the shadowing the property which ensures that pseudo-orbits
of a homeomorphism can be traceable by actual orbits even if rounding errors in
computing are not inevitable, has been recently proved.7;33
4.3. Shadowing, parameter-shifted shadowing and orbit-shifted
shadowing properties
The shadowing property is the property which ensures that pseudo-orbits (i.e. orbits computed numerically using …nite precision number) of a homeomorphism can
be traceable by actual orbit. The concept and primary results of shadowing property for uniformly hyperbolic di¤eomorphisms were introduced by D. V. Anosov34
and R. Bowen,35 who proved that such di¤eomorphisms have the shadowing property. Various papers associated with shadowing property and uniform hyperbolicity
were presented by many authors. Comprehensive expositions on these works were
included in books by K. Palmer36 or S. Y. Pilyugin.37
The precise de…nition of this property is38
De…nition 4.1. (Shadowing) Let (X; d) a metric space, f : X ! X a function and
q
let > 0. A sequence fxk gk=p , (p; q) 2 N2 of points is called a -pseudo-orbit of f if
d (f (xk ) ; xk+1 ) < for p k q 1 (e.g. a numerically computed orbit is a pseudoq
orbit). Given > 0, the pseudo-orbit fxk gk=p , is said
shadowed by x 2 X, if
k
d f (x) ; xk < for p k
q. One says that f has the shadowing property if
for > 0 there is > 0 such that every -pseudo-orbit of f can be
shadowed by
some point.
However, if one relaxes the uniform hyperbolicity condition on di¤eomorphisms
slightly (Lozi map, for example, has not such a property), then many problems concerning the shadowing property are not solved yet. Nowadays, it is widely supposed
that many di¤eomorphisms producing chaos would not have the shadowing
property. In fact, Yuan and Yorke39 found an open set of absolutely nonshadowable C 1 maps for which nontrivial attractors supported by partial hyperbolicity
include at least two hyperbolic periodic orbits whose unstable manifolds have di¤erent dimensions. Moreover, Abdenur and Díaz40 showed recently that the shadowing
property does not hold for di¤eomorphisms in an open and dense subset of the set
of C 1 -robustly nonhyperbolic transitive di¤eomorphisms.
7
)
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In order to …x this issue the notion of the parameter-shifted shadowing property
(for short PSSP ) was introduced by E. M. Coven, I. Kan and J. A. Yorke.41 In
fact, they proved that the tent map (Eq. 12) has the parameter-…xed
shadowing
p
property for almost every sloop a in the open interval I = ( 2; 2); but do not have
it for any a in a certain dense subset of I. Moreover, they proved that, for any
a 2 I, the tent map Ta has PSSP. The de…nition of this variant property being33
De…nition 4.2. (PSSP) Let ffa ga2J be a set of maps on R2 where J is an open
interval in R.
For > 0, the sequence fxn gn>0
R2 is called -pseudo-orbit of fa if
kfa (xn ) xn+1 k
for any integer n > 0.
For a 2 J, we say that fa has the parameter shif ted shadowing property
if, for any > 0, there exist = (a; ) > 0; a
~=a
~ (a; ) 2 J such that any
-pseudo-orbit fxn gn=0 of fa can be -shadowed by an actual orbit of fa~ ,
that is, there exists a y 2 R2 such that kfa~n (y) xn+1 k
for any n > 0.
For Lozi map, S. Kiriki and T. Soma proof that33
Theorem 4.1. There exists a nonempty open set O of the Misiurewicz domain M
such that, for any (a; b) 2 O, the Lozi map La;b has the parameter-shifted shadowing
property in a one-parameter family fLa~;b ga~2J …xing b, where J is a small open
interval containing a.
However they note that “The problem of parameter-…xed shadowability is still
open even for the Lozi family as well as the Hénon family.”
Another variation of shadowability is the orbit-shifted shadowing property (for
short OSSP ) recently introduced by A. Sakurai7 in order to study generalized Lozi
maps introduced by L.-S. Young.42 The limited extend of this article does not allow
us to recall the complex de…nition of these generalized maps.
De…nition 4.3. For 0 > 0, 1 > 0, and > 0, with 0
0 , a sequence fxn gn=0
in R = [0; 1] [0; 1] is an ( 0 ; 1 ) shif ted
pseudo orbit of f if, for any n 1,
xn and xn+1 satisfy the following conditions.
(i) kf (xn ) xn+1 k
if B 0 (xn 1 ) \ (S1 [ S2 ) = ;.
(ii) kf (xn ) ( 1 ; 0) xn+1 k
if B 0 (xn 1 ) \ S1 6= ;.
(iii) kf (xn ) + ( 1 ; 0) xn+1 k
if B 0 (xn 1 ) \ S2 6= ;.
where S1 and S2 are the two components of the essential singularity set of f .
De…nition 4.4. (OSSP) We say that f has the orbit-shifted shadowing property
if, for any > 0 with
> 0 such that any ( 0 ; 1 )
0 , there exist 0 ; 1 ;
shif ted
pseudo orbit fxn gn=0 in R of f can be -shadowed by an actual orbit
of f , that is, there exists z 2 R such that kf n (z) xn k
for any n 0.
Theorem 4.2. Any generalized Lozi map f satisfying the conditions (2.1)–(2.4)
given in7 has the orbit-shifted shadowing property. More precisely, for any 0 <
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> 0 such that any ( ; =2)-shifted -pseudo-orbit of f in R is
0 , there exists
-shadowed by an actual orbit.
For sake of simplicity, we do not give the technical conditions (2.1)-(2.4), however
one can proof that any original Lozi maps La;b indicated in Theorem 4.1 satis…es
these conditions.
In conclusion to both Secs. 3 and 4, it appears that if, owing to the introduction
of computers since forty years, it is easy to compute orbits of mapping in 1 or
2-dimension (and more generally in …nite dimension), it is not unproblematic to
obtain reliable results. The mathematical study of what is actually computed (for
the orbits) is a rather di¢ cult and still challenging problem. Trusty results are
obtained under rather technical assumptions. Considering a new mathematical
tool, the Global Orbit Pattern (GOP ) of a mapping on …nite set, R. Lozi and C.
Fiol19;21 obtain some combinatorial results in 1-dimension.
There is not place in this paper to consider other computations than the computations of orbits. However there is a need for several others reliable numerical results
as the Lyapunov numbers,43 the fractal dimensions (correlation, capacity, Haussdor¤,...) and Lyapunov spectrum ,44 topological entropy,45 extreme value laws.46;47
...The researches are very active in these …elds. One can mention as an example
the relationship between the expected value of the period scales and the correlation
dimension for the case of fractal chaotic attractors of D-dimensional maps.48
Theorem 4.3. The expected value of the period scales with round o¤
m=
D=2
is
(22)
where D is the correlation dimension of the chaotic attractor. The periods m have
substantial statistical ‡uctuation. That is, P (m), the probability that the period is
m, is not strongly peaked around m. This probability is given by
p !
1
8m
F
(23)
P (m) =
m
m
where
F (x) =
r
8
1
r
2
erf
x
p
2
(24)
and erf(x) is de…ned by Eq.16
5. Continuous models
We have highlighted the close relationship between continuous and discrete models
via Poincaré map. We have also pointed out that M. Hénon constructed his model in
order to compute more easily Poincaré map of the Lorenz model which is crunching
too fast for direct numerical computations. It is time to study directly these initial
equations together with the metaphorics ones which that follows naturally: the
Rössler and the Chua equations.
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5.1. Lorenz attractor (1963)
The following non linear system of di¤erential equations was introduced by E.
Lorenza in 1963.1 As a crude model of atmospheric dynamics, these equations led
Lorenz to the discovery of sensitive dependence of initial conditions - an essential
factor of unpredictability in many systems (e. g. meteorology).
8:
(x + y)
<x =
:
y = x y xz
::
z = xy
z
(25)
Numerical simulations for an open neighbourhood of the classical parameter
values
8
(26)
3
suggest that almost all points in phase space tend to a strange attractor the Lorenz
attractor. One can note that the system is invariant under the transformation
= 10;
= 28;
=
S(x; y; z) = ( x; y; z)
(27)
This means that any trajectory that is not itself invariant under S must have a
symmetric “twin trajectory.” For > 1 there are three …xed points: the origin and
the two “twin points”
p
p
C =
(
1);
(
1);
1
For the parameter values generally considered, C have a pair of complex eigenvalues with positive real part, and one real, negative eigenvalue. The origin is a
saddle point with two negative and one positive eigenvalue satisfying
0<
3
<
3
<
1
<
2
Thus the stable manifold of the origin W s (0) is two dimensional and the unstable
manifold of the origin W u (0) is one dimensional. As indicated by M. Hénon5 in his
initial publication the ‡ow contracts volumes at a signi…cant rate (see Sec. 4.1). As
the divergence of the vector …eld is ( + + 1) the volume of a solid at time t
can be expressed as
V (t) = V (0)e
a Edward
( + +1)t
V (0)e
13:7t
Lorenz, born on May, 23, 1917 at west Hartford (Connecticut), dead on April, 16, 2008
at Cambridge (Massachusset), studied the Rayleigh-Bénard problem (i. e. the motion of a ‡ow
heated from below, as an approach of the atmospheric turbulence) using a primitive Royal McBee
LPG-30 computer. He …rst considered series of 12 di¤erential equations when he discovered the
"butter‡y e¤ect". Then he simplied his model from 12 to only three equations. As for Hénon
model, his discovery was made by mistake due to rounding errors.
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Figure 18.
Lorenz attractor.
for the classical parameter values. This means that the ‡ow contracts volumes
almost by a factor one million per time unit which is quite extreme. There appears
to exist a forward invariant open set U containing the origin but bounded away
from C . The set U is a double torus (one with two holes), with its holes centered
around the two excluded …xed points. If one lets ' denote the ‡ow of Eq. 25, there
exists the maximal invariant set
\
A=
' (U; t)
t=0
Due to the strong dissipativity of the ‡ow, the attracting set A must have zero
volume. As indicated by W. Tucker49 , it must also contain the unstable manifold
W u (0) of the origin, which seems to spiral around C in a very complicated non
periodic fashion (see Fig. 18).
In particular A contains the origin itself, and therefore the ‡ow on A can not have
a hyperbolic structure. The reason is that …xed points of the vector …eld generate
discontinuities for the return maps (Poincaré maps), and as a consequence, the
hyperbolic splitting is not continuous. Apart from this, the attracting set appears
to have a strong hyperbolic structure.
5.2. Geometric Lorenz attractor
As it was very di¢ cult to extract rigorous information about the attracting set A
from the di¤erential equations themselves, M. Hénon constructed his famous model
(Sec. 4.1) in order to understand numerically its inner structure. In another way of
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research, a geometric model of the Lorenz ‡ow was introduced by J. Guckenheimer
and R. Williams50 at the end of the seventies (see Fig. 19). This model has been
extensively studied and it is well understood today. The question whether or not
the original Lorenz system for such parameter values has the same structure as the
geometric Lorenz model has been unsolved for more than 30 years. By combination of normal form theory and rigorous computations, W. Tucker51 answered this
question a¢ rmatively, that is, for classical parameters, the original Lorenz system
has a robust strange attractor which is given by the same rules as for the geometric
Lorenz model. From these facts, it is known that the geometric Lorenz model is
crucial in the study of Lorenz dynamical systems. Oddly enough the original equations introduced by Lorenz have remained a puzzle until the same author proved in
1999 in his Ph.D Thesis49 the following theorem
Theorem 5.1. For the classical parameter values, the Lorenz equations support a
robust strange attractor A. Furthermore the ‡ow admits a unique SRB measure X
with supp ( X ) = A.
Remark 5.1. SRB measures, are an invariant measures introduced by Ya. Sinai,
D. Ruelle and R. Bowen in the 1970’s. These objects play an important role in the
ergodic theory of dissipative dynamical systems with chaotic behavior. Roughly
speaking, SRB measures are the invariant measures most compatible with volume
when volume is not preserved. They provide a mechanism for explaining how local
instability on attractors can produce coherent statistics for orbits starting from large
sets in the basin.
In addition, W. Tucker indicates, “In fact, we prove that the attracting set is a
singular hyperbolic attractor. Almost all nearby points separate exponentially fast
until they end up on opposite sides of the attractor. This means that a tiny blob of
initial values soon smears out over the entire attractor. It is perhaps worth pointing
out that the Lorenz attractor does not act quite as the geometric model predicts.
The latter can be reduced to an interval map which is everywhere expanding. This
is not the case for the Lorenz attractor: there are large regions in the attracting set
that are contracted in all directions under the return map. However, such regions
only require a few more iterations before accumulating expansion. This corresponds
to the interval map being eventually expanding, and does not lead to any di¤erent
qualitative long time behaviour. Apart from this, the Lorenz attractor is just as the
geometric model predicts: it contains the origin, and thus has a very complicated
Cantor book structure.”
One of the many ingredients required for the proof of this theorem is a rigourous
arithmetics on a trapping region consisting in 350 adjacent rectangles belonging to
the return plate z = 27 =
1. W. Tucker says “One major advantage of our
numerical method is that we totally eliminate the problem of having to control
the global e¤ects of rounding errors due to the computer’s internal ‡oating point
representation. This is achieved by using a high dimensional analogue of interval
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Figure 19.
Geometric Lorenz model from 52
arithmetic. Each object
(e. g. a rectangle or a tangent vector) subjected to
computation is equipped with a maximal absolute error
, and can thus be
represented as a box
=[ 1
;
+
]
[
; n+
].
1
n
1
1
n
n
When following an object from one intermediate plane to another, we compute
upper bounds on the images of i + i , and lower bounds on the images of i
i,
i = 1;
; n. This results in a new box e
e , which strictly contains the exact
image of
. To ensure that we have strict inclusion, we use quite rough
estimates on the upper and lower bounds. This gives us a margin which is much
larger than any error caused by rounding possibly could be. Hence, the rounding
errors are taken into account in the computed box e
e , and we may continue
to the following intermediate plane by restarting the whole process.
As long as we do not ‡ow close to a …xed point, the local return maps are
well de…ned di¤eomorphisms, and the computer can handle all calculations. Some
rectangles, however, will approach the origin (which is …xed point), and then the
computations must be interrupted as discussed in the previous section (i. e. the
normal form theory).”
The work of W. Tucker is a big step forward reliability of numerical solution of
chaotic continuous dynamical system. However it has been obtained after a great
deal of e¤ort (he published two revised versions of the proof anytime he detected
a mistake in the code) which can not be a¤orded to any new continuous chaotic
model.
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Figure 20.
Lorenz map from.52
5.3. Lorenz map
In contrast the analysis of the Poincaré map associated to the geometric model
of the Lorenz ‡ow allows the use of the parameter-shifted shadowing property.52
This …rst return map on a Poincaré cross section of a geometric Lorenz ‡ow is
called a Lorenz map L : n ! , where
= (x; y) 2 R2 ; jxj ; jyj 6 1 and
= (0; y) 2 R2 ; jyj 6 1 . More precisely it is de…ned as follows
De…nition 5.1. (Lorenz map) Let
denote the components of n with
3
( 1; 0) (see Fig. 19). A map L : n ! , is said to be a Lorenz map if it is a
piecewise C 1 di¤eomorphism which has the form
L (x; y) = ( (x) ; (x; y))
where : [ 1; 1] n f0g ! [ 1; 1] is a piecewise C 1 -map with symmetric property
( x) =
(x) and satisfying
limx!0+ (x) = 1;
limx!0+ 0 (x) = 1;
(1) < 1p
(x) > 2 for any x 2 (0; 1]
0
(see Fig. 20a), and : n ! [ 1; 1] is a contraction in the y-direction. Moreover,
it is required that the images L ( + ), L ( ) are mutually disjoint cusps in , where
the vertices v+ , v of L ( + ) are contained in f 1g [ 1; 1] respectively (see Fig.
20b).
Then, S. Kiriki and T. Soma proved the parameter-shifted shadowing property
(PSSP ) for Lorenz maps.52
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Theorem 5.2. There exists a de…nite set L of Lorenz maps satisfying the following
condition:
For any L 2 L and any > 0, there exist > 0 and > 0 such that any
pseudo orbit of the Lorenz map L with L (x; y) = L (x; y) ( x; 0) is shadowed
by an actual orbit of L.
Theorem 5.3. Any geometric Lorenz ‡ow controlled by a Lorenz map L 2 L has
the parameter-shifted shadowing property.
Due to the lack of space, we refer to52 for the strict description of L.
Besides this work on the Lorenz map, there are many some others rigorous
results on Lorenz equations. Z. Galias and P. Zygliczyński,53;54 for example, are
basing their result on the notion of the topological shifts maps (TS-maps). Let
= (x; y; z) 2 R3 ; z =
1 which is the standard choice for the Poincaré section.
Let P be a Poincaré map generated on the plane by Eq. 25, they prove that:
Theorem 5.4. For all parameter values in a su¢ cently small neighborhood of parameter value given by Eq. 26, there exists a transversal section I
such that
the Poincaré map P induced by Eq. 25 is well de…ned and continuous on I. There
exists a continuous surjective map : Inv I; P2 ! 2 , such that
P2 =
The preimage of any periodic sequence from
2
contains periodic points of P2 .
Remark 5.2. The maximal invariant part of I (with respect to P2 ) is de…ned by
\
Inv I; P2 =
PjI 2i (I)
i2Z
Z
and 2 = f0; 1;
; K 1g is a topological space with the Tichonov topology;
0; 1;
; K 1 being symbols characterizing periodic in…nite sequences of TS-maps
(see4 fore more details).
In this case the Poincaré map is issued directly from Lorenz equations not from
the geometric model. Due to the limited extend of this article, we do not cite all
the results on computer aided proof. We refer the reader to.4
5.4. Rössler attractor (1976)
We have seen that from the seminal discovery of the Lorenz attractor, several strategies have been developped in order to study this appealing and complex mathematical object: geometric Lorenz model, Lorenz map, Hénon map (and Lozi map as
linearized version). In 1976 O. E. Rössler followed a di¤erent direction of research:
instead of simplify the mathematic equation 25 and considering that, due to extreme
simplifaction, there is no actual link between this equation and the Rayleigh-Benard
problem from which it is originate, he turned his mind to the study of a chemical
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Figure 21.
Rössler attractor.
multi-vibrator. He started to design some three-variable oscillator based on a twovariable bistable system coupled to a slowly moving third-variable. The resulting
three-dimensional system was only producing limit cycles at the time. At an international congress on rhythmic functions held on September 8-12, 1975 in Vienna,
he met A. Winfree, a theoretical biologist who challenged him to …nd a biochemical reaction reproducing the Lorenz attractor. Rössler failed to …nd a chemical or
biochemical reaction producing the Lorenz attractor but, he instead found a simpler type of chaos in a paper he wrote during the 1975 Christmas holidays.55 The
obtained chaotic attractor (see Fig. 21)
8:
<x = y z
:
y = x + ay
::
z = b + z(x
(28)
c)
a = 0:2; b = 0:2; c = 5:7
(29)
does not have the rotation symmetry of the Lorenz attractor (de…ned by Eq. 27),
but it is characterized by a map equivalent to the Lorenz map.
The reaction scheme (see Fig. 22) leading to Eq. 28 is meticulously analyzed
by Ch. Letellier and V. Messager.56
The structure of the Rössler attractor is simpler than the Lorenz’s one. However
even if hundreds of papers has been written on it, the rigorous proof of its existence
is not yet established as done for Lorenz equations. In 1997, P. Zygliczyński using
reducted Rössler equations to two parameters instead of three (i. e. stating that
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Figure 22. Combination of an Edelstein switch with a Turing oscillator in a reaction system
producing chaos. E = switching subsystem, T = oscillating subsystem; constant pools (sources
and sinks) have been omitted from the scheme as usual (From,56 adapted from.55 )
a = b in Eq. 28) gave, using computer assisted proof similar results as he did for
4
Lorenz equations.
n
o
Let = (x; y; z) 2 R3 ; x = 0; y < 0; x > 0
Theorem 5.5. For all parameter values in a su¢ cently small neighborhood of
(a; c) = (0:2; 5:7) there exists a transversal section N
such that the Poincaré
map P induced by Eq. 28 is well de…ned and continuous. There exists a continuous
surjective map : Inv (N; P) ! 3 , such that
P=
A
(Inv (N; P)), where
2
01
A = 40 1
10
3
1
15
0
The preimage of any periodic sequence from
A
contains periodic points of P.
As this result is related to the Poincaré map of Rössler equations, nowadays
there is still a need to apply the method developed by W. Tucker to prove the true
existence of a strange attractor.
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5.5. Chua attractor (1983)
Before he discovered his equations, O. Rössler in collaboration with F. Seelig, inspired by a little book entitled Measuring signal generators, Frequency Measuring
Devices and Multivibrators from the Radio-Amateur Library,57 began to “translate”
electronic systems into nonlinear chemical reaction systems. The know-how he developed led eventually in to his chaotic model. Few years later, in october 1983,
visiting T. Matsumoto at Waseda University, L. O. Chua found an electronic circuit
(see Fig. 23) mimicking directly on an oscilloscope screen a chaotic signal (see Fig.
24). As we have seen, only two autonomous systems of ordinary di¤erential equations were generally accepted then as being chaotic, the Lorenz equations and the
Rössler equations. The nonlinearity in both systems is a function of two variables
namely, the product function which is very di¢ cult to build in electronic circuit.
L. O. Chua58 says, “I was to have witnessed a live demonstration of presumably
the world’s …rst successful electronic circuit realization of the Lorenz Equations, on
which Professor Matsumoto’s research group had toiled for over a year. It was indeed a remarkable piece of electronic circuitry. It was painstakingly breadboarded to
near perfection, exposing neatly more than a dozen IC components, and embellished
by almost as many potentiometers and trimmers for …ne tuning and tweaking their
incredibly sensitive circuit board. There would have been no need for inventing a
more robust chaotic circuit had Matsumoto’s Lorenz Circuit worked. It did not.
The fault lies on the dearth of a critical nonlinear IC component with a near-ideal
characteristic and a su¢ ciently large dynamic range; namely, the analog multiplier.
Unfortunately, this component was the key to building an autonomous chaotic circuit in 1983.” He adds, “Prior to 1983, the conspicuous absence of a reproducible
functioning chaotic circuit or system seems to suggest that chaos is a pathological
phenomenon that can exist only in mathematical abstractions, and in computer
simulations of contrived equations. Consequently, electrical engineers in general,
and nonlinear circuit theorists in particular, have heretofore paid little attention to
a phenomenon which many had regarded as an esoteric curiosity. Such was the state
of mind among the nonlinear circuit theory community, circa 1983. Matsumoto’s
Lorenz Circuit was to have turned the tide of indi¤erence among nonlinear circuit
theorists. Viewed from this historical perspective and motivation, the utter disappointments that descended upon all of us on that uneventful October afternoon
was quite understandable. So profound was this failure that the wretched feeling persisted in my subconscious mind till about bedtime that evening. Suddenly
it dawned upon me that since the main mechanism which gives rise to chaos, in
both the Lorenz and the Rössler Equations, is the presence of at least two unstable
equilibrium points - 3 for the Lorenz Equations and 2 for the Rössler Equations
- it seems only prudent to design a simpler and more robust circuit having these
attributes.
Having identi…ed this alternative approach and strategy, it becomes a simple
exercise in elementary nonlinear circuit theory to enumerate systematically all such
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Figure 23.
Chua’s circuit from.61
circuit candidates, of which there were only 8 of them, and then to systematically
eliminate those that, for one reason or another, can not be chaotic.”
The Chua’s equations
8:
(x))
< x = (y
:
(30)
y =x y+z
::
z=
y
(x) = x + g (x) = m1 x +
= 15:60;
1
(m0
2
m1 ) [jx + 1j
= 28:58; m0 =
1
2
; m1 =
7
7
jx
1j]
(31)
(32)
where soon carefully analyzed by T. Matsumoto.59 The main mathematical idea
underneath behind the invention of this circuit is the same one as simplifying the
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Figure 24.
Chua attractor.
Hénon map by linearizing the parabola with an absolute value as done in Lozi map
six years before (even if L. O. Chua did not know these maps at that time).
The “nonlinear” characteristic which is in fact only piecewise linear allows
some exact computations. Henceforth, L. O. Chua et al. proved very soon that
the mechanism of chaos exists in this attractor.60 However in spite it is possible to obtain closed formula for the solution of Eqs. 30 in every subspace
fx
1g ; f 1 < x < 1g ; f1 xg ; due to the transcendental nature of the equation allowing the computation at the matching boundaries fx = 1g ; fx = 1g, of
the global solution,9 it is not possible to compute it explicitely. As only numerically
computed solutions are avaliable, it remains the gnawing problem of what is really
computed. To that problem may be added the fact that the electronic realization
of the Chua’s circuit is not exactly governed by Eqs. 30, due to the instability of
the electronics components, the parameter value (Eq. 32) is randomly ‡uctuating
around its mean value. It is very di¢ cult to analize the experimental observed
chaos.61
One possible way to perform this analysis may be the use of a new mathematical
tool: con…nor instead of attractor.62 The con…nor theory, when applied to Chua’s
circuit allowed the discovery of coexisting chaotic regimes.8;63
6. conclusion
We have shown, in the limited extend of this article, on few but well known
examples,10 that it is very di¢ cult to trust in numerical solution of chaotic dynamical dissipative systems. In some cases one can even proof that it is never possible to
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obtain reliable results. We have focused our survey on models related to the seminal
Lorenz model which is the most studied model of strange attractor. These models
are not the only one studied today. Countless models presenting chaos, arising in
every …elds concerned with dynamics such as ecology, biology, chimistry, economy,
…nance, electronics ..., are developped since forty years. However their study is less
carefully done as those presented here, conducting sometimes to hasty results and
‡awed theories in these sciences. In addition the disturbing phenomenom of “ghost
solution”can appears when discretization of nonlinear di¤erential equation by central di¤erence scheme is used (several examples of such ghost solutions when central
di¤erence scheme is used sole or in combination with Euler’scheme, are given by M.
Yamaguti and al.64 ). Recently A. N. Sharkovsky and S. A. Berezovsky65 pointed
out the notion of “numerical turbulence” which appears, due to incorrectness of
calculation method stipulated by discreteness. In conclusion one can say that there
is room for more study of the relationship between numerical computation and
theoretical behavior of chaotic solutions of dynamical systems.
References
1. E. N. Lorenz, “Deterministic nonperiodic ‡ow,” J. Atmospheric Science, 20, 130-141
(1963).
2. M. Misiurewicz, “Strange attractors for the Lozi mappings,” Ann. New-York Acad.
Sci., 357 (1), 348-358 (1980).
3. W. Tucker, “The Lorenz attractor exists,” C. R. Acad. Sci. Paris Sér. I Math., 328
(12), 1197-1202 (1999).
4. P. Zygliczyński, “Computer assisted proof of chaos in the Rössler equations and the
Hénon map,” Nonlinearity, 10 (1), 243-252 (1997).
5. M. Hénon, “A Two-dimensional mapping with a strange attractor,” Commun. Math.
Phys., 50, 69-77 (1976).
6. R. Lozi, “Giga-periodic orbits for weakly coupled tent and logistic discretized maps,”
Modern Mathematical Models, Methods and Algorithms for Real World Systems, eds.
A. H. Siddiqi, I. S. Du¤ & O. Christensen, (Anamaya Publishers, New Delhi, India),
80-124 (2006).
7. A. Sakurai, “Orbit shifted shadowing property of generalized Lozi maps,” Taiwanese
J. Math., 14 (4), 1609-1621 (2010).
8. R. Lozi & S. Ushiki, “Coexisting chaotic attractors in Chua’s circuit,” Int. J. Bifurcation & Chaos, 1 (4), 923-926 (1991).
9. R. Lozi & S. Ushiki, “The theory of con…nors in Chua’s circuit: accurate analysis of
bifurcations and attractors,” Int. J. Bifurcation & Chaos, 3 (2), 333-361 (1993).
10. D. Blackmore, “New models for chaotic dynamics,” Regular & Chaotic dynamics, 10
(3), 307-321 (2005).
11. J. Laskar & P. Robutel, “The chaotic obliquity of the planets,” Nature, 361, 608-612
(1993).
12. Z. Elhadj & J. C. Sprott, “A two-dimensional discrete mapping with C 1 multifold
chaotic attractors,” Elec. J. Theoretical Phys., 5 (17), 1-14 (2008).
13. P.-F. Verhulst, “Notice sur la loi que la population poursuit dans son accroissement,”
Correspondance mathématique et physique de l’observatoire de Bruxelles, 4 (10), 113121 (1838).
14. P.-F. Verhulst, “Recherches mathématiques sur la loi d’accroissement de la popu-
March 24, 2012
20:24
Proceedings Trim Size: 9.75in x 6.5in
swp0000
39
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
lation,” Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres de
Bruxelles, 18, 1-42 (1845).
R. M. May, Stability and Complexity of Models Ecosystems, Princeton University
Press, Princeton, NJ (1973).
R. M. May,“Biological populations with nonoverlapping generations: stable points,
stable cycles, and chaos,” Science, New Series, 186 (4164), 645-647 (1974).
M. J. Feigenbaum, “The universal metric properties of nonlinear transformations,” J.
Stat. Phys., 21, 669-706 (1979).
S. M. Ulam & J. von Neumann, “On combination of stochastic and deterministic
processes,” Bulletin Amer. Math. Soc., 53 (11), 1120 (1947).
R. Lozi & C. Fiol, “Global orbit patterns for dynamical systems on …nite sets,” Conference Proceedings Amer. Inst. Phys., 1146, 303-331 (2009).
O. E. Lanford III, “Some informal remarks on the orbit structure of discrete approximations to chaotic maps,” Experimental Mathematics, 7 (4), 317-324 (1998).
R. Lozi & C. Fiol, “Global orbit patterns for one dimensional dynamical systems,”
Grazer Math. Bericht., 354, 112-144 (2009).
R. Lozi, “Complexity leads to randomness in chaotic systems,”Mathematical Methods,
Models, and Algorithms in Science and Technology, eds. A. H. Siddiqi, R. C. Singh
and P. Manchanda, (World Scienti…c, Singapore), 93-125 (2011).
M. S. Baptista, “Cryptography with chaos,” Phys. Lett. A, 240, 50-54 (1998).
M. R. K. Ari¢ n & M. S. M. Noorani, “Modi…ed Baptista type chaotic cryptosystem
via matrix secret key,” Phys. Lett. A, 372, 5427-5430 (2008).
K. T. Alligood, T. D. Sauer and J. A. Yorke, Chaos. An introduction to dynamical
systems, Springer, Textbooks in mathematical sciences, New-York (1996).
A. N. Sharkovski¼¬, “Coexistence of cycles of a continuous map of the line into itself,”
Int. J. Bifurcation & Chaos, 5 (5), 1263-1273 (1995).
J. C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, Oxford, UK
(2003).
P. Diamond, P. Kloeden, A. Pokrovskii, and A. Vladimorov, “Collapsing e¤ects in
numerical simulation of a class of chaotic dynamical systems and random mappings
with a single attracting centre,” Physica D, 86, 559-571 (1995).
P. Diamond & A. Pokrovskii, “Statistical laws for computational collapse of discretized
chaotic mappings,” Int. J. Bifurcation & Chaos, 6 (12A), 2389-2399 (1996).
G. Yuan & J. A. Yorke, “Collapsing of chaos in one dimensional maps,” Physica D,
136, 18-30 (2000).
D. Ruelle & F. Takens, “On the nature of turbulence,” Comm. Math. Phys., 20,
167-192 (1971).
R. Lozi, “Un attracteur étrange du type attracteur de Hénon,” J. Physique, 39 (C5),
9-10 (1978).
S. Kiriki & T. Soma, “Parameter-shifted shadowing property of Lozi maps,” Dyn.
Sys., 22 (3), 351-363 (2007).
D. V. Anosov, “Geodesic ‡ows on closed Riemann manifolds with negative curvature,” Proc. Steklov Math. Inst., 90 (1967). Amer. Math. Soc. Trans., Providence,
R.I. (1969).
R. Bowen, “!-limit sets for Axiom A di¤eomorphisms,” J. Di¤ . Eq., 18, 333-339
(1975).
K. Palmer, Shadowing in Dynamical Sytems: Theory and Appplications, Kluwer Academic Publications (2000).
S. Y. Pilyugin, “Shadowing in Dynamical Systems,”Lecture Notes in Math., Springer,
1706, (1999).
March 24, 2012
20:24
Proceedings Trim Size: 9.75in x 6.5in
swp0000
40
38. K. Sakai, “Di¤eomorphisms with the shadowing property,” J. Austral. Math. Soc.
(Series A), 61, 396-399 (1996).
39. G.-C. Yuan & J. A. Yorke, “An open set of maps for which every point is absolutely
nonshadowable,” Proc. Amer. Math. Soc., 128, 909-918 (2000).
40. F. Abdenur & L. J. Díaz, “Pseudo-orbit shadowing in the C 1 topology,” Discrete and
Continuous Dynamical Systems, 17 (2), 223-245 (2007).
41. E. M. Coven, I. Kan and J. A. Yorke, “Pseudo-orbit shadowing in the family of tent
maps,” Trans. Amer. Math. Soc., 308, 227-241 (1988).
42. L.-S. Young, “A Bowen-Ruelle measure for certain piecewise hyperbolic maps,”Trans.
Amer. Math. Soc., 287, 41-48 (1985).
43. D. Delenau, “Dynamic Lyapunov indicator: a practical tool for distinguishing between
ordered and chaotic orbits in discrete dynamical systems,”Proc. 13th WSEAS Intern.
Conf. on Math. methods, eds. N. Gavriluta, R. Raducanu, M. Iliescu, H. Costin, N.
Mastorakis, Iasi, Romania, 117-122 (2011).
44. P. Grassberger, “ On Lyapunov and dimension spectra of 2D attractors, with an
application to the Lozi map” J. Phys. A: Math. Gen., 22, 585-590 (1989).
45. Y. Ishii & D. Sands, “Monotonicity of the Lozi family near the tent-maps,” Comm.
Math. Phys., 198, 397-406 (1998).
46. M. P. Holland, R. Vitolo, P. Rabassa, A. E. Sterk, and H. W. Broer, “Extreme value
laws in dynamical systems under physical observables,” Physica D, 241 (5), 497–513
(2012).
47. D. Faranda, V. Lucarini, G. Turchetti, and S. Vaienti, “Extreme Value distribution
for singular measures,” arXiv:1106.2299, to appear on Chaos (2012).
48. C. Grebogi, E. Ott, and J. A. Yorke, “Roundo¤-induced periodicity and the correlation
dimension of chaotic attractors,” Phys. Rev. A, 38 (7), 3688-3692 (1988).
49. W. Tucker, “The Lorenz attractor exists,”Ph.D thesis, Uppsala Univ., Sweden, (1999).
50. J. Guckenheimer & R. F. Williams, “Structural stability of Lorenz attractors,” Publ.
Math. I.H.E.S., 50, 59-72 (1979).
51. W. Tucker, “A rigorous ODE solver and Smale’s 14th problem,” Found. Comput.
Math., 2, 53-117 (2002).
52. S. Kiriki & T. Soma, “Parameter-shifted shadowing property for geometric Lorenz
attractors,” Trans. Amer. Math. Soc., 357 (4), 1325-1339 (2004).
53. Z. Galias & P. Zygliczyński, “Chaos in the Lorenz equations for classical parameter values a computer assisted proof,” Universitatis Iagellonicae Acta Mathematica,
XXXVI, 209-210 (1998).
54. Z. Galias & P. Zygliczyński, “Computer assisted proof of chaos in the Lorenz system,”
Physica D, 115, 165-188 (1995).
55. O. E. Rössler, “Chaotic behavior in simple reaction system,” Zeitschrift für Naturforschung, A 31, 259–264 (1976).
56. Ch. Letellier & V. Messager, “In‡uences on Otto Rössler’s earliest paper on chaos,”
Int. J. Bifurcation & Chaos, 20 (11), 1-32 (2010).
57. H. Sutaner, Me sender Frequenzmesser un Multivibratoren, Radio-Praktiker Bucherei,
128/130, Franzis Verlag, Munich, (1966).
58. L. O. Chua, “The genesis of Chua’s Circuit,” AEÜ, 46 (4), 250-257 (1992).
59. T. Matsumoto,“A chaotic attractor from Chua’s circuit,”IEEE Trans., CAS-31 (12),
1055-1058 (1984).
60. L. O. Chua, M., Kumoro, and T. Matsumoto, “The Double Scroll Family,” IEEE
Trans. Circuit and Systems, 32 (11), 1055-1058 (1984).
61. L. O. Chua, L. Kocarev, K. Eckert, and M. Itoh, “Experimental chaos synchronization
in Chua’s circuit,” Int. J. Bifurcation & Chaos, 2 (3), 705-708 (1992).
March 24, 2012
20:24
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swp0000
41
62. R. Lozi & S. Ushiki, “Con…nors and bounded-time patterns in Chua’s circuit and the
double-scroll family,” Int. J. Bifurcation & Chaos, 1 (1), 119-138 (1991).
63. S. Boughaba & R. Lozi, “Fitting trapping regions for Chua’s attractor. A novel method
based on Isochronic lines,” Int. J. Bifurcation & Chaos, 10 (1), 205–225 (2000).
64. M. Yamaguti & S. Ushiki, “Chaos in numerical analysis of ordinary di¤erential equations,” Physica D, 3 (3), 618-626 (1981).
65. A. N. Sharkovsky & S. A. Berezovsky “Phase transitions in correct-incorrect calculations for some evolution problems,” Int. J. Bifurcation & Chaos, 13 (7), 1811-1821
(2003).